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My objective function is $\log_2(1+{x^2y^2})$ and I found two upper bounds for $x^2$ and $y^2$.

For example, assumed that we have the following upper bounds: $x^2\leq\text{constant}_1^2$ and $y^2\leq\text{constant}_2^2$.

So, my question is: could we say that $\log_2(1+{x^2y^2})\leq \log_2(1+{\text{constant}_1^2\cdot\text{constant}_2^2})$? If so, how tight is this bound?

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  • $\begingroup$ I would start with using $x^2y^2$ as objective. $\endgroup$ Apr 11, 2021 at 19:37
  • $\begingroup$ Actually, the main objective function is $\sum_{i}\log_2(1+x_i^2y_i^2)$ $\endgroup$ Apr 12, 2021 at 9:44

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Yes, because $\log$ is monotonic, it preserves inequalities. The tightness depends on your other constraints.

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